Fatuma Kassim, a mother-of-seven, said: "We have no hope now and I think this is the end of Mogadishu."
Aid agencies say that about one million Somalis live as internal refugees.
Residents said that there was heavy shelling overnight as AU forces clashed with armed men opposed to the Somali government, as well as the presence of Ethiopian and AU forces.
A group calling itself the Mujahidins of Raskamboni said that attacked one of the peacekeepers' bases sparking the fighting.
"This is the heaviest fighting ever since the AU deployed. I have seen the African Union forces using tanks"
It was a retaliatory attack against the African forces and it was the heaviest ever waged against them," Mohamoud Dulyadeyn, a spokesman, said.
He said that his group "operates in Somali territories carrying out attacks against the enemy of Allah".
At least seven civilians were killed by artillery fire in two districts of southern Mogadishu, witnesses said.
"This is the heaviest fighting ever since the AU deployed. I have seen the African Union forces using tanks," Farah Hassan, a Mogadishu resident, said.
"I have seen many civilians crowded in minibuses [leaving the battle zone] and there was no access to hospital for the wounded in the whole neighbourhood."
Residents of the Taleh district told AFP that the shelling shattered many residential houses and set fire to a number of businesses.
In another development, Ethiopian fficials and the Paris-based Medecins du Monde said on Tuesday that two foreign aid workers kidnapped in Ethiopia have been taken to central Somalia.
Local Somali authorities have sent security officials to seek the hostages' freedom from the unidentified abductors.
"We have sent security forces to search for the aid workers, who were brought to the region late yesterday," Ali Sheikh Hashi, a local official, said.
The pair, both employees of the French aid agency, were seized from Fadhigaradle village where they were visiting drought-hit areas in Ethiopia on Monday, Hareri Hassan Barre, the commissioner for the Balanbale district in central Somalia, said.
Medecins du Monde confirmed the abductions, but did not give the nationality of its employees.
"The organisation is in permanent contact with the authorities, its team on the ground as well as other actors in the field," it said in a statement.
Somali fighters freed on Monday a German national and his Somali wife who had been abducted over the weekend in the northern Somali breakaway state of Puntland, where kidnapping is endemic.
Kidnappers have also been holding three journalists - a Canadian, an Australian and a Somali - since August 23 and are reportedly demanding $2.5 million for their release.
The fighting in Somalia came after gun battles had broken out in several areas of Mogadishu over the weekend following an attack on the international airport as an African Union aircraft was landing. Scores of people have been killed in recent days.
Anti-government fighters appear to have gained strength in recent weeks with the al-Shabaab armed group, an offshoot of the Islamic Courts' Union, which controlled much of Somalia in 2006, attacking AU bases and removing government checkpoints.
Sheikh Muktar Roboow, a spokesman for the movement, has vowed that attacks against the 2,000 AU peacekeepers, many of whom are based at the airport, will be intensified
"We are going to double our attacks against the AU forces. The only option they have is to leave our country," he said.
The troops were deployed in March 2007 to help Abdullahi Yusuf Ahmed, the president, bring the nation under the government's control.